{
  "abstract": "We study the finite element approximation of the Dirichlet problem for the Bessel $(p,s)$-Laplacian, $\\operatorname{div}^{s} \\left(|\\nabla^{s} u|^{p-2}\\nabla^{s} u\\right)$, built from the Riesz fractional gradient $\\nabla^s$ and posed on the Bessel potential space $X^{s,p}$, obtained by complex interpolation between $L^p$ and $W^{1,p}$. For $p = 2$ this operator reduces to the fractional Laplacian, while for $p \\neq 2$ it differs from the fractional $p$-Laplacian arising from real interpolation. Since a direct Galerkin discretization requires reassembling a dense stiffness matrix at every nonlinear iteration, we propose an augmented Lagrangian formulation based on the projected fractional gradient $B_h = \\mathbfΠ_h \\nabla^s$, in which the only dense matrix is assembled once and the nonlinearity decouples elementwise. We analyze the resulting consistency error, establish a priori convergence rates in $X^{s,p}$, conditional on a discrete inf-sup condition for the augmented Lagrangian scheme, and present numerical experiments that illustrate the convergence of the method and its ability to deal with degenerate and strongly nonlinear problems.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.37747",
  "author_total": 2,
  "authors": [
    {
      "name": "Juan Pablo Borthagaray"
    },
    {
      "name": "José Camilo Rueda Niño"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fa67y32w9j4ze.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.37747",
  "fulltext_url": null,
  "id": "fa67y32w9j4ze",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.37747",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "2206|Advanced Numerical Methods in Computational Mathematics"
    ]
  ],
  "openalex": "W7214945431",
  "pdf_url": "https://arxiv.org/pdf/2609.37747",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fa67y32w9j4ze.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.37747",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.37747"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Finite element approximation for the Bessel $(p,s)$-Laplacian",
  "title_text": "Finite element approximation for the Bessel (p,s)-Laplacian",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/fa67y32w9j4ze",
  "venue": "arXiv",
  "versions": []
}